In Harish-Chandra’s classification of the discrete series representations for $G(\mathbf R)$, there are some unusual aspects of the formulas for their characters. Langlands confronted this in the work that led to his classification. With the principle of functoriality in his mind, Langlands eventually proposed the theory of endoscopy. And then Shelstad made an endoscopic extension of Langlands’ classification. Here are some great references:


 - For classical results of semisimple $\mathbf R$-groups due to Harish-Chandra, Knapp‘s book ***Representation Theory of Semisimple Groups*** covers almost everything you need to know.

 - For the endoscopic extension, it seems that articles written by Clozel, Adams and other people in the collection ***Stabilization of the Trace Formula, Shimura Varieties, and Arithmetic Applications*** perfectly meets your requirements. Actually Shelsta herself wrote a lot of notes on such topics, you can find them on her website.

 - For general results and some examples on the forms of real groups, see Chapter III of Serre’s book ***Cohomologie Galoisienne***.